Topics in Algebraic Topology and its Applications I
This is a series of courses on selected important topics in algebraic topology, which aims to help the research of the graduate students and junior researchers in pure and applied algebraic topology. During this semester, our selected topics will concentrate on Leray-Serre spectral sequence and its relevant topics, consisting of Barratt-Gugenheim-Moore simplicial fibre bundles theory, simplicial approach to the Leray-Serre spectral sequence with its comparison with the classical approach in the textbooks, sampling computations of spectral sequences performed by participated students, the applications in computing homotopy groups, Witten’s twisted de Rham cohomology with its comparison with Brown’s twisted tensor products, Grigor’yan-Muranov-Yau’s delta-cohomology and Delta-twisted homology.
Lecturer
Date
10th September ~ 31st December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 17:05 - 20:55 | A3-4-301 | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Algebraic Topology
Syllabus
1. Homology with local coefficients
2. Exact couples
3. Homology of fibre spaces: Elementary Theory
4. Homology of fibre spaces
5. Computations of homology using Leray-Serre spectral sequence by examples
6. Brown twisted tensor, Witten’s twisted de Rham cohomology
7. Simplicial fibration/fibre bundles, twisted homology
8. Computations of homology using Leray-Serre spectral sequence or twisted (co-)homology by examples
2. Exact couples
3. Homology of fibre spaces: Elementary Theory
4. Homology of fibre spaces
5. Computations of homology using Leray-Serre spectral sequence by examples
6. Brown twisted tensor, Witten’s twisted de Rham cohomology
7. Simplicial fibration/fibre bundles, twisted homology
8. Computations of homology using Leray-Serre spectral sequence or twisted (co-)homology by examples
Reference
1. George W. Whitehead, Elements of Homotopy Theory, Springer New York, NY eBook ISBN 978-1-4612-6318-0 published 2012
2. Robert M. Switzer, Algebraic Topology - Homotopy and Homology, Springer Berlin, Heidelberg. eBook ISBN 978-3-642-61923-6 published 2017
3. John McCleary, A User's Guide to Spectral Sequences, Cambridge University Press, 2000 ISBN-13 978-0521567596
2. Robert M. Switzer, Algebraic Topology - Homotopy and Homology, Springer Berlin, Heidelberg. eBook ISBN 978-3-642-61923-6 published 2017
3. John McCleary, A User's Guide to Spectral Sequences, Cambridge University Press, 2000 ISBN-13 978-0521567596
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
No
Language
Chinese
Lecturer Intro
Jie Wu received a Ph.D. degree in Mathematics from the University of Rochester and worked as a postdoc at Mathematical Sciences Research Institute (MSRI), University of California, Berkeley. He was a former tenured professor at the Department of Mathematics, National University of Singapore. In December 2021, he joined the Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA). His research interests are algebraic topology and applied topology. His main achievements in algebraic topology are to establish the fundamental relations between homotopy groups and the theory of braids, and the fundamental relations between loop spaces and modular representation theory of symmetric groups. In applied topology, he has obtained various important results on topological approaches to data science. Currently, he leads his research team for bi-directional applications of topology aiming to topological foundation of high-order interactions in complex systems, developing algebraic topology of directed graphs and hypergraphs, and applying topological approaches beyond persistent homology to practical problems. He has published more than 100 academic papers in top journals such as “the Journal of American Mathematical Society”, “Proceedings of the National Academy of Sciences of the United States of America”,“Advances in Mathematics”, etc. In 2007, he won the "Singapore National Science Award”. In 2014, his project was funded by the “Overseas Joint Fund of National Natural Science Foundation” (Jieqing B).